Module categories, internal bimodules, and Tambara modules

Abstract

We use the theory of Tambara modules to extend and generalize the reconstruction theorem for module categories over a rigid monoidal category to the nonrigid case. We show a biequivalence between the 2‐category of cyclic module categories over a monoidal category and the bicategory of algebra and bimodule objects in the category of Tambara modules on . Using it, we prove that a cyclic module category can be reconstructed as the category of certain free module objects in the category of Tambara modules on , and give a sufficient condition for its reconstructability as module objects in . To that end, we extend the definition of the Cayley functor to the nonclosed case, and show that Tambara modules give a proarrow equipment for ‐module categories, in which ‐module functors are characterized as 1‐morphisms admitting a right adjoint. Finally, we show that the 2‐category of all ‐module categories embeds into the 2‐category of categories enriched in Tambara modules on , giving an “action via enrichment” result.